Space like strong unique continuation for sublinear parabolic equations

Banerjee, Agnid; Manna, Ramesh

doi:10.1112/jlms.12317

Cited by 5 publications

(4 citation statements)

References 42 publications

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“…In this work we generalise Rüland's result to degenerate elliptic equations of the type (2.26). For related results in the parabolic setting, we refer to [3] and [11]. We have the following generalization of the result of Rüland for the variable coefficient Baouendi-Grushin operators.…”

Section: 1mentioning

confidence: 99%

Carleman estimates for a class of variable coefficient degenerate elliptic operators with applications to unique continuation

Banerjee¹,

Manna²

2020

Preprint

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In this paper, we obtain new Carleman estimates for a class of variable coefficient degenerate elliptic operators whose constant coefficient model at one point is the so called Baouendi-Grushin operator. This generalizes the results obtained by the two of us with Garofalo in [9] where similar estimates were established for the "constant coefficient" Baouendi-Grushin operator. Consequently, we obtain: (i) a Bourgain-Kenig type quantitative uniqueness result in the variable coefficient setting; (ii) and a strong unique continuation property for a class of degenerate sublinear equations. We also derive a subelliptic version of a scaling critical Carleman estimate proven by Regbaoui in the Euclidean setting using which we deduce a new unique continuation result in the case of scaling critical Hardy type potentials.

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Section: 1mentioning

confidence: 99%

Carleman estimates for a class of variable coefficient degenerate elliptic operators with applications to unique continuation

Banerjee¹,

Manna²

2020

Preprint

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“…For a differential operator P on a domain Ω, the strong unique continuation property (abbreviated to sucp in what follows) for |P u| ≤ |V u| means that a nontrivial solution u to |P u| ≤ |V u| cannot vanish to infinite order (in a suitable sense) at any point. The sucp for second order parabolic operator has been studied by many authors (see [22,27,25,4,6,8,7,9,19,1] and references therein). In particular, the study of sucp for the heat operator with time-dependent potentials goes back to Poon [25] and Chen [4], who considered bounded potentials.…”

Section: Introductionmentioning

confidence: 99%

Unique continuation for the heat operator with potentials in weak spaces

Jeong¹,

Lee²,

Ryu³

2021

Preprint

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We prove strong unique continuation property for the differential inequalityIn particular, we establish the strong unique continuation property for V ∈, which has been left open since the works of Escauriaza [6] and Escauriaza-Vega [8]. Our results are consequences of the Carleman estimates for the heat operator in the Lorentz spaces.The sucp for the Laplacian −∆ is better understood. Since the pioneering work of Carleman [3], most of subsequent results were obtained by following his idea, the Carleman weighted inequality. In particular, Jerison and Kenig [13] proved the sucp for the Laplacian with V ∈ L d/2 loc , d ≥ 3. Their result was extended by Stein [28] to potentials V ∈ L d/2,∞ under the assumption that V L d/2,∞ is small enough. Later, Wolff [33] showed that the smallness assumption is indispensable if V ∈ L d/2,∞ . Here, • p,r denotes the norm of the Lorentz space L p,r (R d ) (for example, see [29]).

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“…In this work we generalise Rüland's result to degenerate elliptic equations of the type (1.11). For related results in the parabolic setting, we refer to [4] and [12]. We have the following generalization of the result of Rüland for the variable coefficient Baouendi-Grushin operators.…”

mentioning

confidence: 99%

“…Throughout the paper we assume that 12) where I N indicates the identity matrix in R N . In order to state our main assumptions (H) on the matrix A it will be useful to represent the latter in the following block form…”

mentioning

confidence: 99%

Carleman estimates for a class of variable coefficient degenerate elliptic operators with applications to unique continuation

Banerjee¹,

Manna²

2021

DCDS

Self Cite

View full text Add to dashboard Cite

In this paper, we obtain new Carleman estimates for a class of variable coefficient degenerate elliptic operators whose constant coefficient model at one point is the so called Baouendi-Grushin operator. This generalizes the results obtained by the two of us with Garofalo in [10] where similar estimates were established for the "constant coefficient" Baouendi-Grushin operator. Consequently, we obtain: (i) a Bourgain-Kenig type quantitative uniqueness result in the variable coefficient setting; (ii) and a strong unique continuation property for a class of degenerate sublinear equations. We also derive a subelliptic version of a scaling critical Carleman estimate proven by Regbaoui in the Euclidean setting using which we deduce a new unique continuation result in the case of scaling critical Hardy type potentials.

show abstract

Space like strong unique continuation for sublinear parabolic equations

Cited by 5 publications

References 42 publications

Carleman estimates for a class of variable coefficient degenerate elliptic operators with applications to unique continuation

Carleman estimates for a class of variable coefficient degenerate elliptic operators with applications to unique continuation

Unique continuation for the heat operator with potentials in weak spaces

Carleman estimates for a class of variable coefficient degenerate elliptic operators with applications to unique continuation

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